Metrical irrationality results related to values of the Riemann $\zeta$-function

TL;DR

This paper introduces a family of series related to the Riemann zeta function and proves their values at integers are linearly independent over rationals for almost all parameters, providing new criteria for irrationality of key constants.

Contribution

It establishes linear independence results for a new family of series associated with the zeta function and derives novel irrationality criteria for important mathematical constants.

Findings

Values at integers are linearly independent for almost all parameters.

New criteria for the irrationality of zeta constants and Euler--Mascheroni constant.

Results extend to series involving exponential and factorial terms.

Abstract

We introduce a one-parameter family of series associated to the Riemann $\zeta$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the parameter, where almost all is with respect to any sufficiently nice measure. We also give similar results for the Euler--Mascheroni constant, for $\sum_{n=1}^\infty \frac{1}{n^n}$ and for $\sum_{n=1}^\infty \frac{1}{n! +1}$. Finally, specialising the criteria used, we give some new criteria for the irrationality of $\zeta(k)$, the Euler--Mascheroni constant and the latter two series.